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Computers and Geotechnics 38 (2011) 721–730
Contents lists available at ScienceDirect
Computers and Geotechnics
journal homepage: www.elsevier .com/locate /compgeo
Effective medium methods and a computational approach for
estimatinggeomaterial properties of porous materials with randomly
oriented ellipsoidal pores
A.P. Suvorov ⇑, A.P.S. SelvaduraiDepartment of Civil Engineering
and Applied Mechanics, McGill University, Montreal, QC, Canada H3A
2K6
a r t i c l e i n f o a b s t r a c t
Article history:Received 12 November 2010Received in revised
form 25 February 2011Accepted 8 April 2011Available online 11 May
2011
Keywords:Effective elastic propertiesDisc-shaped
voidsSelf-consistent estimates of elasticpropertiesMori–Tanaka
estimates of elastic propertiesComputational estimates
0266-352X/$ - see front matter � 2011 Elsevier Ltd.
Adoi:10.1016/j.compgeo.2011.04.002
⇑ Corresponding author.E-mail addresses: [email protected]
(A.P. S
mcgill.ca (A.P.S. Selvadurai).
The role of effective medium approaches and the differential
scheme in estimating the overall elasticmoduli of a porous medium
with randomly oriented pores is examined. The analytical estimates
of theelastic moduli are compared with approximations available in
the literature that are valid for small poreaspect ratios. Accuracy
of the analytical estimates is further established by performing
finite elementsimulations. Finite element estimates are obtained
for a model of a porous medium containing three fam-ilies of
spheroidal pores arranged in a mutually orthogonal configuration.
This model is regarded as aclose approximation to a porous medium
with randomly oriented pores. Also, experimental data avail-able
for several sandstones and granites was used to analytically
examine the influence of the aspect ratioof pores on overall
properties. This information is also used to provide an estimate
for the permeability ofthe porous medium.
� 2011 Elsevier Ltd. All rights reserved.
1. Introduction
This work relates to the determination of the overall
elasticmoduli of a porous medium. The pores of the region can be
emptyor fluid-filled, and are treated as either discrete spherical
voids oras oblate spheroidal voids with small aspect ratios. The
distribu-tion of pore orientations is assumed to be random. On the
macro-scale such a porous medium is considered to be
statisticallyhomogeneous and isotropic, i.e., there are two overall
elastic mod-uli (e.g., bulk and shear moduli) that characterize the
elastic behav-ior of the material.
While exact analytical expressions for the overall elastic
moduliof a porous material are difficult to determine, several
proceduresexist for estimating the elastic moduli of such a solid.
Among themare the popular effective medium methods that include the
self-consistent method [15] and the Mori–Tanaka method [18]. The
dif-ferential scheme [17] is also a popular method for obtaining
theoverall properties [29]. For flat oblate spheroidal pores with
smallaspect ratio (i.e., penny-shaped cracks), Kachanov’s scheme is
fre-quently used to estimate the overall elasticity properties
[16].
The works of Benson et al. [2] and Gueguen and Sarout [12]have
shown that Kachanov’s estimates for the overall elastic mod-uli of
a cracked material provide a good fit to the experimentaldata for
several rocks. Bary [1] used the Mori–Tanaka method toestimate the
elastic moduli of porous cement paste. Giraud et al.
ll rights reserved.
uvorov), patrick.selvadurai@
[8] estimated the elastic moduli of rocks consisting of
flattenedellipsoidal pores with an aspect ratio of 1/20 and
spherical inclu-sions of mineral phases using the Mori–Tanaka
method. Gruescuet al. [10] used a two-step homogenization procedure
to deter-mine the overall thermal conductivity of porous rocks
(i.e., theMori–Tanaka method to find the properties of clay with
calciteand quartz inclusions and the method of Ponte Castaneda and
Wil-lis [21] to account for the presence of pores in the clay
matrix). Theaspect ratio of the oblate spheroidal pores used by
Gruescu et al.[10] was 1/20. Giraud, Gruescu et al. [9,10] used the
Mori–Tanakamethod for two stages of the homogenization procedure to
deter-mine the overall thermal conductivity of porous rocks. Guery
et al.[13] estimated the elastic properties of an argillite
composed of aclay matrix and spherical inclusions of quartz and
calcite usingthree approaches: the dilute, the self-consistent and
the Mori–Tanaka methods. They found that the Mori–Tanaka method
wasable to accurately reproduce the experimental data for the
elasticmoduli.
Roberts and Garboczi [23] used a finite element model of aporous
medium with overlapping spherical pores and overlappingoblate
spheroidal pores with an aspect ratio of 1/4. They found thatthe
differential scheme accurately matched their finite elementresults
for the overall Young’s modulus for porosities of up to0.5. The
self-consistent estimates were only accurate for porositieslower
than 0.2. Whether the differential scheme or the self-consistent
method can be used to accurately estimate the overallelastic
properties of a porous medium with the cracks of a smalleraspect
ratio cannot be decided a priori.
http://dx.doi.org/10.1016/j.compgeo.2011.04.002mailto:[email protected]:patrick.selvadurai@
mcgill.camailto:patrick.selvadurai@
mcgill.cahttp://dx.doi.org/10.1016/j.compgeo.2011.04.002http://www.sciencedirect.com/science/journal/0266352Xhttp://www.elsevier.com/locate/compgeo
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722 A.P. Suvorov, A.P.S. Selvadurai / Computers and Geotechnics
38 (2011) 721–730
A finite element model of the porous material consisting of
ran-domly distributed penny-shaped cracks with small aspect
ratioscan be used to computationally assess which of the estimates
forthe overall moduli yields the most accurate result. The
computa-tional approach, while formally straightforward, does not
alwayslead to convenient implementation since the mesh should
conformto the complex geometry of the matrix-pore interfaces.
Computa-tional experience with the use of multiphysics finite
element codeCOMSOL™ suggests that only a model having a relatively
smallnumber of cracks, i.e., porous media with very small values
ofporosities, can be successfully meshed and analyzed. Even for
verysmall porosities, the size of the representative volume
elementmust be sufficiently small to ensure a successful generation
ofthe required finite element mesh.
The program COMSOL™ has been successfully used for solvingfluid
flow problems for the real 3D pore geometry of a granularmaterial
obtained through X-ray computer assisted micro-tomog-raphy [19].
The microstructure of the material consisted of solidgrains of
approximately spherical shape and a pore-network do-main with a
volume fraction of 0.35.
2. Estimates of overall elastic moduli and their properties
Consider a porous medium with randomly-distributed pores ofany
shape. The pores can be empty (dry) or fluid-filled. (The
elasticconstants of the matrix are Ks, Gs, Es, ms, the bulk modulus
of thefluid is Kf.) The porosity or volume fraction of the pores is
denotedby n. For such a porous medium some important relationships
canbe summarized.
Consider the constitutive equation for the isotropic stress rI
inthe porous medium under undrained conditions [4,22,20,24,25],
rI ¼ KD�V � ap ¼ Ku�V ð1Þ
where �V is the volumetric strain, p is the fluid pressure, KD
is thebulk modulus of the porous medium with empty pores
(drainedbulk modulus), Ku is the bulk modulus of the porous medium
withfluid-filled pores (undrained bulk modulus), a = 1�KD/Ks is the
Biotcoefficient. For a porous medium under undrained conditions, a
fur-ther constitutive relation is,
a�V þnKfþ a� n
Ks
� �p ¼ 0 ð2Þ
From (1) and (2) we can write the Biot–Gassmann relationship
as,
Ku ¼ KD þa2
n=Kf þ ða� nÞ=Ksð3Þ
This equation relates the bulk modulus of the porous medium
withfluid-filled pores Ku to the bulk modulus of the porous medium
withempty pores KD.
Consider now the constitutive equation for the deviatoric
stres-ses sij in the porous skeleton. Since the fluid in the pores
is assumedto be inviscid, the deviatoric stress is,
sij ¼ 2GDeij ¼ 2Gueij ð4Þ
where GD is the shear modulus of the porous medium with
emptypores (drained shear modulus), Gu is the shear modulus of the
por-ous medium with fluid-filled pores (undrained shear
modulus).From (4) it is clear that the shear moduli of the drained
and un-drained porous medium satisfy the relationship,
Gu ¼ GD ð5Þ
Experimental data presented by Thomsen [27] for several
rockssuggests that the equalities (3) and (5) must be indeed
satisfied. Itturns out that not all estimates for the effective
moduli of the por-
ous medium satisfy the relationships (3) and (5). To
demonstratethis we restrict our attention to the porous medium with
ellipsoi-dal pores.
Consider first a porous medium in which pores have the shapeof
oblate spheroids with a small aspect ratio, q� 1 (q is the ratioof
the semi-minor axis to the semi-major axis of the spheroid).When q�
1, these correspond to penny-shaped cracks, but forbrevity we shall
refer to these pores as cracks. The cracks are as-sumed to be
randomly distributed.
Mori–Tanaka estimates for the overall elastic moduli of such
aporous material with empty pores are [3],
KD ¼ Ks 1þ g169
1� m2s1� 2ms
� ��1
GD ¼ Gs 1þ g32 1� msð Þð5� msÞ
45ð2� msÞ
� ��1ð6Þ
where g = 3n/(4pq) is the crack density, and n is the porosity.
Theseequations have been derived with the assumption that the
aspectratio of the oblate spheroids is much smaller than unity,
i.e., inthe limit q ? 0. Thus, they are essentially a truncated
version ofMori–Tanaka estimates valid only for small aspect
ratios.
For a porous material with penny-shaped cracks, the schemeby
Kachanov can also be used to estimate the components ofthe overall
compliance matrix. For a random distribution ofempty cracks,
components of the overall compliance matrix takethe form [12],
S1111¼1=Esþ32ð1�m2s Þ3ð2�msÞEs
g3�ms
2g5
� �¼1=Esþg
32ð1�m2s Þ3ð2�msÞEs
10�3ms30
S1122¼ S1133¼�msEsþ 32ð1�m
2s Þ
3ð2�msÞEs�ms
2g
15
� �¼�ms
Es1þg16ð1�m
2s Þ
45ð2�msÞ
� �
S1212¼1=ð4GDÞ¼1=ð4GsÞþ32ð1�m2s Þ3ð2�msÞEs
14
2g3�ms
2g
15
� �ð7Þ
The bulk modulus KD can be obtained from (7) asKD = (S1111 +
S1122 + S1133)�1/3, and the overall shear modulus asGD = 1/4S1212.
This will lead to the results of the Mori–Tanaka meth-od (6). Thus,
for an isotropic distribution of empty cracks, Kacha-nov’s scheme
is equivalent to the truncated version of the Mori–Tanaka method
derived under the limit q ? 0.
Consider now a porous medium with randomly oriented fluid-filled
cracks. Kachanov’s estimate of the components of the
overallcompliance matrix are,
S1111 ¼ 1=Es þ32ð1� m2s Þ3ð2� msÞEs
g3þ wg
5
� �
S1122 ¼ S1133 ¼ �msEsþ
32 1� m2s� �
3ð2� msÞEsw
g15
� �
S1212 ¼ 1=ð4GDÞ ¼ 1=ð4GsÞ þ32ð1� m2s Þ3ð2� msÞEs
14
2g3þ w g
15
� �ð8Þ
where
w ¼ 1� ms2
� � df1þ df
� 1; df ¼3ð1� 2msÞpq
4ð1� m2s ÞKsKf� 1
� �ð9Þ
Upon simplifications and using the identities, 1/KD = 3(S1111 +
S1122 + S1133), 1/GD = 4S1212, we obtain expressionsfor the bulk
and shear moduli of the porous medium with fluid-filled cracks
(undrained elastic moduli), as follows:
Ku ¼ Ks 1þ g169
1� m2s1� 2ms
df1þ df
� ��1
Gu ¼ Gs 1þ g32ð1� msÞ 3þ
df1þdfð2� msÞ
� �45ð2� msÞ
0@
1A�1
ð10Þ
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Table 1Normalized bulk and shear moduli, Ku/Ks, Gu/Gs, for
porous medium with randomlyoriented fluid-filled pores estimated by
Kachanov’s scheme (K), non-truncatedversion of Mori–Tanaka method
(MT), effective medium method 1 (EM1), self-consistent method (SC),
differential scheme (DIF). Porosity is 0.05.
Ku/Ks
Aspect ratio, q K MT EM1 SC DIF1 – 0.9017 0.8994 0.8971
0.89951/100 0.6812 0.6801 0.6792 0.6608 0.67251/200 0.6703 0.6698
0.6692 0.6587 0.66371/300 0.6665 0.6662 0.6658 0.6587 0.66121/1000
0.6611 0.6610 0.6608 0.6587 0.6588
Gu/Gs1 – 0.9087 0.9066 0.9042 0.90671/100 0.4748 0.4494 0.4379
0.0703 0.32631/200 0.3167 0.3001 0.2900 0.0 0.12921/300 0.2377
0.2254 0.2169 0.0 0.05291/1000 0.0866 0.0822 0.0785 0.0 0.0001
Table 2Normalized bulk and shear moduli, KD/Ks, GD/Gs, for
porous medium with randomlyoriented empty pores estimated by
Kachanov’s scheme (K), non-truncated version ofMori–Tanaka method
(MT), effective medium method 1 (EM1), self-consistentmethod (SC),
differential scheme (DIF). Porosity is 0.05.
KD/Ks
Aspect ratio K MT EM1 SC DIF1 – 0.8786 0.8752 0.8717 0.87531/100
0.1716 0.1643 0.1575 0.000 0.04721/200 0.0938 0.0896 0.0855 0.000
0.00481/300 0.0646 0.0615 0.0587 0.000 0.00051/1000 0.0203 0.0193
0.0184 0.000 0.0000
GD/Gs1 – 0.9087 0.9066 0.9042 0.90661/100 0.3784 0.3590 0.3476
0.000 0.13221/200 0.2333 0.2215 0.2129 0.000 0.01521/300 0.1687
0.1602 0.1534 0.000 0.00171/1000 0.0574 0.0545 0.0519 0.000
0.0000
Fig. 1. Normalized overall moduli for porous medium under
undrained conditionsestimated by non-truncated version of
Mori–Tanaka method (thick solid line),Kachanov’s scheme (thin solid
line), self-consistent method (dash-dot line) anddifferential
scheme (dashed line).
A.P. Suvorov, A.P.S. Selvadurai / Computers and Geotechnics 38
(2011) 721–730 723
If the bulk modulus of the fluid Kf ? 0, df ?1 and (10) reduces
tothe results for the porous medium with empty cracks given by
(6).We observe, however, that as predicted by Kachanov’s theory,
theshear modulus of the porous medium with fluid-filled cracks Gu
isnot equal to GD.
The Kachanov’s estimates for the overall bulk moduli KD and
Kusatisfy Biot–Gassmann relationship (3). To justify this
statement,note that from (61) and the definition of Biot
coefficient,34 ð1� 2msÞ= 1� m2s
� �¼ 43 gð1� aÞ=a, where g = 3n/(4pq) is the crack
density. Then, from definition of df in (9) it follows that df =
n(Ks/Kf�1)(1 � a)/a. Using this expression for the df in (101) it
can beshown that the Biot–Gassmann relationship is satisfied.
The Mori–Tanaka estimates also satisfy relationship (3) for
anyshape of the randomly-distributed pores, i.e., with any aspect
ratio.The justification for this statement is given in Appendix A
(see alsoTables 1 and 2). Thus, the equivalence of the estimates of
the over-all drained bulk modulus KD, proven previously, implies a
similarequivalence of the estimates of the undrained bulk modulus
Ku.
3. Numerical results and modelling
Table 1 and Fig. 1 give the estimates of the normalized
overallbulk and shear moduli for the porous medium with
fluid-filledcracks and spherical cavities (i.e., Ku/Ks, Gu/Gs). The
estimates areobtained by Kachanov’s scheme (K), various effective
mediummethods including the self-consistent method (SC) and the
differ-ential scheme (DIF). The Mori–Tanaka (MT) estimates are not
trun-cated (i.e., they can be used for any aspect ratio). The
estimates of
effective medium method 1 (EM1) are obtained by choosing
theelastic moduli of the comparison (reference) medium as (1 �
n)Ks, (1 � n)Gs. Porosity n = 0.05, Ks = 25 GPa, ms = 0.3, Kf = 2.2
GPa.The results given in Table 1 and Fig. 1 indicate that
Kachanov’s esti-mates for the overall moduli of the porous medium
with penny-shaped cracks are the closest to the Mori–Tanaka
estimates andconverge to the Mori–Tanaka estimates faster than to
any otherestimate as the aspect ratio q ? 0.
Table 2 and Fig. 2 give the estimates of the normalized
overallbulk and shear moduli of the porous medium with empty, or
dry,cracks and spheres, i.e., KD/Ks, GD/Gs. The estimates are
obtainedby Kachanov’s scheme (K), the non-truncated version of
theMori–Tanaka method (MT), the self-consistent method (SC),
effec-tive medium method 1, Budiansky and O’Connell method [5],
andthe differential scheme. The estimates of effective medium
method1 (EM1) are obtained by choosing the elastic moduli of the
compar-ison (reference) medium as (1 � n)Ks, (1 � n) Gs. The
porosity is setto 0.05.
Fig. 3 compares the difference between the shear moduli Gu
andits Biot-consistent counterpart GBCu ¼ GD. The smallest
difference be-tween the shear moduli Gu and GD is obtained from
Kachanov’s
-
Fig. 2. Normalized overall moduli for porous medium under
drained conditionsestimated by non-truncated version of Mori–Tanaka
method (thick solid line),Kachanov’s scheme (thin solid line),
self-consistent method (thick dash-dot line),Budiansky and
O’Connell method (thin dash-dot line), and differential
scheme(dashed line).
Fig. 3. Normalized difference between undrained moduli estimated
by non-truncated version of Mori–Tanaka method, Kachanov’s scheme,
self-consistentmethod or differential scheme and the respective
Biot-consistent moduli obtainedfrom the equalities (3) and (5).
Table 3Poroelastic moduli for various sandstones and granites as
given by Zimmerman [30].Aspect ratio q of penny-shaped pores
(cracks) is evaluated by Mori–Tanaka method
GD (GPa) KD (GPa) a n q g ¼ 3n4pq
Berea sandstone 6 8 0.79 0.19 1/13.3 0.6033Weber sandstone 12 13
0.64 0.06 1/35.6 0.5099Ohio sandstone 6.8 8.4 0.74 0.19 1/12.4
0.5625Pecos sandstone 5.9 6.7 0.83 0.20 1/18.4 0.8785Boise
sandstone 4.2 4.6 0.85 0.26 1/15.9 0.9869Ruhr sandstone 13 13 0.65
0.02 1/123.6 0.5901
Tennessee marble 24 40 0.19 0.02 1/13.1 0.0625Charcoal granite
19 35 0.27 0.02 1/18.4 0.0879Westerly granite 15 25 0.47 0.01
1/82.5 0.1970
724 A.P. Suvorov, A.P.S. Selvadurai / Computers and Geotechnics
38 (2011) 721–730
scheme and the Mori–Tanaka method, and the largest
differenceoccurs in the self-consistent method. However, for the
porousmedium with spherical pores and disks the relationship Gu =
GDis satisfied by all estimates. The same conclusion can be
drawnfrom Tables 1 and 2.
Fig. 3 also shows the difference between the bulk moduli Ku
andits Biot-consistent counterpart KBCu . The latter is evaluated
from Eq.(3) using the estimated value of the bulk modulus KD. For
all poreaspect ratios, the Kachanov’s and Mori–Tanaka estimates for
theoverall bulk modulus satisfy Biot–Gassmann relationship (3).
Theestimates obtained by effective medium method 1 also
satisfyBiot–Gassmann relationship (3) as it can be inferred from
Tables1 and 2 (not shown in the figure). However, the
self-consistentand differential scheme estimates do not. For the
porous mediumwith spherical pores and disks, Biot–Gassmann
relationship (3) issatisfied by all estimates.
Table 3 shows experimentally determined properties of
severalsandstones and granites documented by Zimmerman [30]. Also,
itshows their porosities and Biot coefficients a. Using the
Mori–Ta-naka method, the aspect ratio of the pores was
back-calculated
by using the optimization toolbox available in MatLabTM.
Theobjective function was the difference between the elastic
moduliof the drained material determined using the Mori–Tanaka
method
-
Fig. 4. The finite element model of a cuboidal region of a
porous material with theoblate spheroidal pores of aspect ratio
1/10. The volume fraction of pores is 0.0361.
A.P. Suvorov, A.P.S. Selvadurai / Computers and Geotechnics 38
(2011) 721–730 725
and the experimental values of the moduli, i.e.,KMTD � K
expD
� �2þ GMTD � G
expD
� �2. The input for MatLab™ built-in
optimization function ‘fminimax’ was the set of
experimentallymeasured elastic moduli KexpD , G
expD , and the Biot coefficient a
exp.The result obtained was the shear modulus of the matrix
materialGs and aspect ratio of the pores q. The optimal solution
corre-sponded to the zero value of the objective function.
From Table 3 it can be seen that the aspect ratio of
pores(cracks) is smaller for rocks with smaller porosities. This is
to beexpected because only in the presence of pores with small
aspectratios can a rock have a large Biot coefficient a (e.g.,
0.65, whenthe porosity is very small). In addition, the pores with
very smallaspect ratios have a low percolation threshold and their
presencecan explain non-zero values of permeability even when the
poros-ity is very small. It should also be noted that the crack
densitiesg = 3n/(4pq) for sandstones listed in Table 3 range from
0.5 to 1but the crack densities for granites and marble are in the
range0.05–0.2. To simplify computations, the rock was assumed to
havepores of a single shape, i.e., a distribution of pore shapes
was notconsidered. The aspect ratio presented in Table 3 should,
therefore,be treated as the average aspect ratio. The presence of
pores withlarge aspect ratios, i.e., spherical pores, is still
possible in the rocktypes examined, but their volume fraction must
be limited and thepresence of a certain volume of flat pores with a
small aspect ratiois necessary. Using information on the average
aspect ratio ofpores, other quantities such as the crack density,
permeabilityand the local stress fields at the crack tip can be
estimated.
We now present the results of the finite element
computations.The computational code COMSOL™ was used to create
severalmodels of the porous medium with orthogonally oriented
butotherwise randomly distributed pores, with the possibility of
theirintersection. In each model all pores have the same size and
aspectratio. The aspect ratios of the pores are 1 (spherical), 1/10
and 1/30.The modelled region has a cuboidal shape with sides five
times lar-ger than the diameter of the spheroidal pores.
The ability of the COMSOL™ code to create such a model
andgenerate the finite element mesh depends on the size of the
model,the volume fraction of the pores, their aspect ratio and
their orien-tation. For spherical pores, the COMSOL™ code was able
to create amodel of the porous medium with porosities as high as
0.2126, butfor pores with an aspect ratio of 1/10 the maximum
porosity thatcould be modelled was only 0.0361. The error messages
generatedby the code while creating a mesh prevent the user from
modellinga material region with higher values of porosities when
the poreshave a small aspect ratio. This is regarded as a
limitation of thecode.
A typical model used in computations is shown in Figs. 4 and
5.The aspect ratio of pores is 1/10, the volume fraction of pores
is0.0361. To simplify the model, instead of randomly oriented
pores,only three families of mutually orthogonal cracks (pores)
weremodelled. It is important to note that the resulting system has
a cu-bic material symmetry. If L is a 6 � 6 stiffness matrix for
this sys-tem, then it is found that L44 < (L11 � L12)/2 whereas
for theisotropic case L44 = (L11 � L12)/2 = G. But for small
porosities andsufficiently large aspect ratio of pores, L44 � (L11
� L12)/2 and thesystem can be considered approximately isotropic.
For the modelshown in Figs. 4 and 5 there are 32 pores oriented
parallel to thex–y plane, 29 pores parallel to the y–z plane, and
28 pores parallelto the x–z plane.
Fig. 6 shows the Biot coefficient a = 1 � KD/Ks estimated by
theMori–Tanaka method (solid line) and the differential
scheme(dashed line) as a function of porosity n. The results of the
finiteelement computations for the porous material with randomly
ori-ented overlapping pores are indicated with stars and open
circles.The elastic properties of the material of the solid phase
are taken asKs = 25 GPa, ms = 0.3.
The Biot coefficient evaluated using the computational schemeis
shown for two situations where the external boundary condi-tions
correspond to either uniform stress or uniform strain. Dueto the
relatively small size of the finite element model employed,the Biot
coefficient depends on the type of the boundary conditionsused.
When the stress boundary condition is applied, the RVE re-sponse is
too compliant and an upper bound for the Biot coefficientis
obtained, i.e., the overall elastic modulus KD turns out to
besmall. When the strain boundary condition is applied, the RVE
re-sponse is too stiff and the lower bound for the Biot coefficient
isobtained, i.e., the overall elastic modulus KD is large. The
correctsolution should lie between these finite element (FE)
bounds.
-
Fig. 5. The side views of the finite element model of a cuboidal
region of a porousmaterial with the oblate spheroidal pores of
aspect ratio 1/10. The volume fractionof pores is 0.0361.
0 0.05 0.1 0.15 0.2 0.25 0.30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
porosity n
Biot
coe
ffici
ent α
= (1
−KD
/Ks)
differential schemeFEM − uniform stressFEM − uniform strain
Mori−Tanaka method
aspect ratio 1
aspect ratio 1/10
aspect ratio 1/30
Fig. 6. The Biot coefficient for a porous material with randomly
distributedoverlapping pores estimated by the Mori–Tanaka method,
the differential schemeand a finite element approach.
0 0.05 0.1 0.15 0.2 0.25 0.30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
porosity n
Shea
r coe
ffici
ent (
1−G
D/G
s)
Mori−Tanaka methoddifferential schemeFEM − uniform stressFEM −
uniform strain
aspect ratio 1
aspect ratio 1/10
aspect ratio 1/30
Fig. 7. Shear coefficient 1 � GD/Gs for a porous material with
randomly distributedoverlapping pores estimated by the Mori–Tanaka
method, the differential schemeand a finite element approach.
726 A.P. Suvorov, A.P.S. Selvadurai / Computers and Geotechnics
38 (2011) 721–730
Fig. 6 shows that for RVE with spherical pores, both
Mori–Tana-ka and differential schemes provide the estimates for the
Biot coef-ficient that lie within the finite element (FE) bounds
although theMori–Tanaka estimate is almost coincident with the
lower bound.For RVE with the pores of aspect ratio 1/10, the
Mori–Tanaka anddifferential scheme estimates are still close to
each other but thedifferential scheme tends to slightly
underestimate the bulk mod-ulus KD because it lies slightly above
the upper FE bound on theBiot coefficient.
Fig. 7 shows the shear coefficient defined as 1 � GD/Gs
estimatedby the Mori–Tanaka method (solid line) and the
differential
scheme (dashed line). The computational results are denoted
bythe stars and circles. The shear coefficient evaluated using the
com-putational approach is shown for both uniform shear stress
anduniform shear strain boundary conditions. The shear coefficient
isagain dependent on the type of the boundary conditions
applied.The imposed uniform shear stress condition results in a
smalleroverall shear modulus GD, and a larger shear coefficient
than thatfor the imposed shear strain condition. Fig. 7 shows that
for thespherical pores, the Mori–Tanaka and differential scheme
esti-mates for the shear coefficient are close to each other and
bothtend to lie closer to the lower finite element bound on the
shearcoefficient. For the pores with aspect ratio 1/10, the
estimates ob-tained from the Mori–Tanaka and differential scheme
are closeonly for very small range of porosities, and for these
porositiesthe Mori–Tanaka estimate falls slightly below the lower
computa-tionally derived bound for the shear coefficient. The small
values ofporosities used in these computations, however, limit
validity ofthese conclusions.
-
Table 4Permeability (m2) for various sandstones and granites as
given by Zimmerman [30]and estimated from Eq. (16). The crack
aperture is 0.5 lm.
n Measured Estimated
Berea sandstone 0.19 1.87E�13 1.51E�15Weber sandstone 0.06
9.86E�16 4.77E�16Ohio sandstone 0.19 5.52E�15 1.51E�15Pecos
sandstone 0.20 7.89E�16 1.59E�15Boise sandstone 0.26 7.89E�13
2.07E�15Ruhr sandstone 0.02 1.97E�16 1.59E�16
Tennessee marble 0.02 9.87E�20 0Charcoal granite 0.02 9.87E�20
0Westerly granite 0.01 3.95E�19 4.17E�18
A.P. Suvorov, A.P.S. Selvadurai / Computers and Geotechnics 38
(2011) 721–730 727
4. New Biot-consistent estimates of elastic moduli
Let K, G be the estimates of the bulk and shear moduli of
theporous medium (drained or undrained). Let us denote the
elasticmoduli of the comparison medium by K0 and G0. Recall that
inthe Mori–Tanaka method K0 = Ks and G0 = Gs, where Ks, Gs are
theelastic moduli of the solid phase. The Mori–Tanaka estimatesKMT,
GMT for two-phase medium play an important role in that
theycoincide with the upper Hashin–Shtrikman bound [14], i.e.,
anymeaningful estimates must satisfy the inequalities,
0 6 K 6 KMT ; 0 6 G 6 GMT ð11Þ
The equalities (11) are satisfied by the effective medium
methods aslong as the elastic moduli of the comparison medium
satisfy the fol-lowing inequalities [28,6]:
0 6 K0 6 maxfKs;Kf g; 0 6 G0 6 Gs ð12Þ
For the drained porous medium the fluid bulk modulus Kf = 0,
forthe undrained one it is not zero but usually much smaller than
Ks.Clearly there are infinite number of K0, G0 that satisfy (12)
and,therefore, infinite number of the estimates can be generated
thatsatisfy the bounds (11). Our goal is, however, to select the
estimatesthat are Biot-consistent, i.e., satisfy both relationships
(3) and (5).
The results presented in Tables 1 and 2, namely those
ofMori–Tanaka method and EM1 method, suggest that if
comparisonmedium is chosen to be identical for drained and
undrained porousmedia, then the Biot–Gassmann relationship (3) is
indeed satisfied.However, the equality of the shear moduli (5) is
satisfied only if thepore shape is spherical or disk-shaped, i.e.,
the pore aspect ratio iseither one or zero.
This suggests that the new Biot-consistent estimates KBC, GBC
forthe porous medium can be defined as follows. Select the
propertiesof the comparison medium for the drained material such
that theinequalities (12) are satisfied with Kf = 0. Given the
properties ofthe comparison material, obtain the estimates of the
bulk andshear moduli for the drained porous medium, i.e., KD, GD.
Let Ku,Gu be the estimates of the elastic moduli for the undrained
porousmedium obtained using the same comparison material. These
esti-mates satisfy the respective Hashin–Shtrikman bounds, i.e.,Ku
6 KMTu , Gu 6 G
MTu as long as Kf 6 Ks which is usually the case.
The estimates KD, Ku are Biot-consistent since they satisfy (3)
but,in general, GD – Gu. Now define the new Biot-consistent
estimatesfor the undrained porous medium simply as,
KBCu ¼ KD þa2
n=Kf þ ða� nÞ=Ks¼ Ku; a ¼ 1�
KDKs
GBCu ¼ GD ð13Þ
The estimate of the bulk modulus KBCu in (13) satisfies the
Hashin–Shtrikman bound (11) simply because the estimate Ku
satisfies thisbound. Furthermore, GBCu satisfies the
Hashin–Shtrikman boundsince GBCu ¼ GD 6 G
MTD 6 G
MTu . Thus, the new estimates defined in
(13) are Biot-consistent and satisfy the Hashin–Shtrikman
bounds.It should be noted that if mD is the Poisson’s ratio for the
drainedporous medium with elastic moduli KD, GD, then the Poisson’s
ratiofor the undrained system with Biot-consistent elastic moduli
KBCu ,GBCu satisfies mD 6 mBCu 6 0:5.
The proposed estimates can be considered a generalization ofthe
results obtained by Thomsen [27] and Endres [7]. Thomsen[27]
modified the self-consistent estimate, namely, the
expressionderived by Budiansky and O’Connell [5] in such a way that
the newestimate is Biot-consistent, i.e., both equalities (3) and
(5) are sat-isfied. To achieve this, the properties of the
comparison mediumwere assumed dependent on the fluid pressure
conditions, i.e.,whether the porous medium is drained or undrained.
Endres [7]used essentially Mori–Tanaka method (called in his paper
by EIAS
method) and modified the concentration factor for the shear
strainso that the new estimate satisfies the equality for the shear
moduli(5).
5. Estimate for permeability
It is natural to ask whether the effective medium
approachesproposed here can be extended to provide estimates for
the perme-ability of a porous medium, which is an important
property of afluid-saturated multiphase media [25]. The
permeability of a por-ous rock containing cracks can be expressed
as [2],
k ¼ 215
fw2qg ð14Þ
where q and g are the aspect ratio and crack density as before,
f isthe percolation factor – fraction of connected pores, and w is
theaverage crack aperture. In the studies by Benson et al. [2] the
crackaperture ranges from 0.2 lm to 1 lm. According to the model
of[11] the percolation factor is equal to zero for small crack
densitiessatisfying p2g/4 < 1/3 and approximately 1 for large
crack densitiessatisfying p2g/4 > 1. For the range 1/3 <
p2g/4 < 1 the percolationfactor can be approximated by the
expression [11],
f ¼ 94
p2
4g� 1
3
� �2ð15Þ
Using definition of the crack density, Eq. (14) can be
simplifiedto,
k ¼ n10p
fw2 ð16Þ
where n is the porosity.Table 4 compares the measured values of
permeability [30] and
the values computed using (16). The average crack aperture
wasassumed to be equal to 0.5 lm for all rocks. It can be seen from
Ta-ble 4 that for sandstones the agreement between measured
andestimated values of the permeability is acceptable at least
whenthe porosity n is sufficiently small. This is perhaps explained
bythe narrower distribution of pore shapes in the low porosity
rocks.When the porosity n is large (e.g., 0.26), the estimate for
the per-meability may be inaccurate but can be corrected by
choosingthe crack aperture to be correspondingly larger than 0.5
lm.
For charcoal granite and marble, the estimated permeability
iszero for the two rocks due to the fact that the percolation
factorf = 0 for small values of crack densities, g = 0.0625,
0.0879. It ap-pears that more refined theories are needed to
estimate the trans-port properties of very low permeability rocks.
The estimate for thepermeability of Westerly Granite with a crack
density of g = 0.1970can also be corrected by making the crack
aperture smaller than0.5 lm.
When the distribution of the pore space is amorphous, the
por-ous medium has a spatial heterogeneity that requires
alternative
-
728 A.P. Suvorov, A.P.S. Selvadurai / Computers and Geotechnics
38 (2011) 721–730
approaches for the estimation of effective properties. In a
recentstudy, Selvadurai and Selvadurai [26] investigated the
effectivepermeability of a porous medium with a spatial
heterogeneity inthe isotropic measure of the point-wise
permeability. Theseauthors have demonstrated through experimental
and computa-tional modelling of the heterogeneity that a geometric
mean pro-vides an accurate estimate for the effective isotropic
permeabilityof a porous medium with spatial heterogeneity of
permeability,which has a lognormal distribution.
6. Conclusions
In this study analytical estimates of the overall elastic moduli
ofa porous medium with randomly oriented flat oblate
spheroidalpores were obtained using effective medium methods and
Kacha-nov’s scheme. It was shown that Kachanov’s scheme is
equivalentto the truncated version of the Mori–Tanaka method when
the as-pect ratio of the pores is very small. It was also
demonstrated thatestimates obtained from the Mori–Tanaka method
satisfy the Biot–Gassmann Eq. (3) for any shape of pores, but the
estimates ob-tained by the self-consistent method or differential
scheme satisfythe Biot–Gassmann equation only for spherical and
disk-shapedpores.
The analytical estimates for the elastic moduli were
comparedwith the results obtained using a computational approach
appliedto a model of a porous medium with overlapping spherical
poresand one containing three families of overlapping oblate
spheroidalpores of identical size arranged in a mutually orthogonal
configu-ration. The computational code COMSOL™ was used to
obtainnumerical results. The ability of the finite element program
COM-SOL™ to create such a model and generate the finite element
meshdepends on the volume fraction of pores and their aspect ratio.
Ingeneral, the smaller the aspect ratio of the pores, the lower
themaximum porosity that can be modelled by using the COMSOL™code
is.
The analytical estimates for the overall bulk modulus agree
clo-sely with the bulk modulus obtained using the computational
ap-proach. The analytical estimates for the overall shear modulus
tendto be closer to the computational estimates obtained using the
uni-form shear strain boundary conditions applied to cuboidal
RVE.However, in order to provide more definitive conclusions,
higherporosities need to be modelled by alternative
computationalstrategies.
The pore aspect ratios for several sandstones and granites
wereestimated using the Mori–Tanaka method and available
experi-mental results. In general, it was found that the aspect
ratio ofpores is smaller for rocks with smaller porosities. The
crack densityof sandstones was found to be higher than the crack
density ofgranites. Using information on the aspect ratio of pores,
the esti-mates for the permeability were also obtained and compared
withthe experimental values. The agreement between the two sets
ofresults was found acceptable for all sandstones with small
porosi-ties, n < 0.06, and even for certain sandstones with
porosities ashigh as n = 0.19, 0.2.
Acknowledgement
The work described in this paper was supported by an
NSERCDiscovery Grant awarded to A.P.S. Selvadurai.
Appendix A
The effective medium methods for estimating the overall
elasticmoduli of the two-phase heterogeneous solids are reviewed in
thisAppendix. All effective medium methods are formally
equivalent
[6]. They differ only in the choice of the comparison
(reference)medium. For example, in the Mori–Tanaka method the
stiffnessof the comparison medium L0 is equal to that of the matrix
mate-rial Ls, and in the self-consistent method L0 = L, where L is
the over-all stiffness matrix of the porous medium that is
sought.
In all effective medium methods, each inclusion or a pore
isenvisioned to be embedded within a large volume of the
compar-ison medium. Thus, the interaction problem between
multipleinclusions is replaced by a much simpler problem of a
single inclu-sion in a homogeneous comparison medium subjected to a
uni-form strain or stress field. If a uniform strain field �0 is
applied tothe comparison medium, the strain field with the
inclusion canbe found as,
�p ¼ Tp�0 ðA:1Þ
where Tp is the strain concentration factor for the inclusion
(orpore). It can be found as,
Tp ¼ ðI � PðL0 � LpÞÞ�1 ðA:2Þ
where P is the Hill’s polarization tensor [28], Lp is the
stiffness ma-trix of the inclusion. The matrix P depends only on
the properties ofthe comparison medium, i.e., L0 and the shape of
the inclusion. Foran isotropic comparison medium, the fourth-order
tensor P is givenby Ponte Castaneda and Willis [21]. We recast
their results in a ma-trix form, i.e., by treating P as a 6 � 6
matrix rather than the tensor.
Suppose that the inclusion is an oblate spheroid that has
theminor semi-axis directed along the x1-axis of the Cartesian
coordi-nate system (x1, x2, x3). In this coordinate system the 6 �
6 matrix Pcan be found as,
P ¼
n0 l0 l0 0 0 0
l0 k0 þm0=4 k0 �m0=4 0 0 0
l0 k0 �m0=4 k0 þm0=4 0 0 0
0 0 0 m0 0 0
0 0 0 0 p0 0
0 0 0 0 0 p0
0BBBBBBBBBBBB@
1CCCCCCCCCCCCA
ðA:3Þ
where n0, l0, k0, m0, p0 are the constants [21] defined in terms
of theaspect ratio of the spheroid q and the elastic moduli K0, G0
of thecomparison medium,
k0 ¼7h� 2q2 � 4q2h�
G0 þ 3K0 h� 2q2 þ 2q2h�
8ð1� q2ÞG0ð4G0 þ 3K0Þ
l0 ¼�ðG0 þ 3K0Þ h� 2q2 þ 2q2h
� 4ð1� q2ÞG0ð4G0 þ 3K0Þ
n0 ¼6� 5h� 8q2 þ 8q2h�
G0 þ 3K0 h� 2q2 þ 2q2h�
2ð1� q2ÞG0ð4G0 þ 3K0Þ
m0 ¼15h� 2q2 � 12q2h�
G0 þ 3K0 3h� 2q2�
16ð1� q2ÞG0ð4G0 þ 3K0Þ
p0 ¼2 4� 3h� 2q2�
G0 þ 3K0 2� 3hþ 2q2 � 3q2h�
8ð1� q2ÞG0ð4G0 þ 3K0ÞðA:4Þ
where for oblate spheroids q < 1,
h ¼q cos�1ðqÞ � qð1� q2Þ1=2h i
ð1� q2Þ3=2
and for prolate spheroids q > 1,
h ¼q �cosh�1ðqÞ þ qðq2 � 1Þ1=2h i
ðq2 � 1Þ3=2
The concentration factor for the matrix phase Ts can be
definedas,
-
A.P. Suvorov, A.P.S. Selvadurai / Computers and Geotechnics 38
(2011) 721–730 729
T s ¼ ðI � PðL0 � LsÞÞ�1 ðA:5Þ
where Ls is the stiffness of the matrix phase, and P is an
isotropicmatrix which can be found from the equality,
fTpg ¼ fðI � PðL0 � LpÞÞ�1g ¼ ðI � PðL0 � LpÞÞ�1 ðA:6Þ
Here double brackets signify averaging over all possible
orienta-tions of the pores. Thus, to determine P the concentration
factorfor the inclusions or pores (A.2) needs to be evaluated and
averagedover all orientations. Note that the formula (A.5), which
gives theconcentration factor for the matrix phase, is rather new
since theuse of arbitrary comparison material to estimate the
properties ofa porous medium with pores of arbitrary shape has not
receivedsufficient attention.
For a random distribution of pores, the components of the 6 �
6concentration factor {T} can be expressed in terms of componentsof
the matrix T as,
fTg11¼3
15ðT11þT22þT33Þþ
115ðT12þT21þT13þT31þT23þT32Þ
þ 215ðT44þT55þT66Þ
fTg12¼1
15ðT11þT22þT33Þþ
215ðT12þT21þT13þT31þT23þT32Þ
� 115ðT44þT55þT66Þ
fTg44¼2
15ðT11þT22þT33Þ�
115ðT12þT21þT13þT31þT23þT32Þ
þ15ðT44þT55þT66Þ
The matrix T can be found in the local coordinate system of
aninclusion (pore) according to (A.2).
The strain in the inclusion can also be expressed in terms of
theoverall strain � applied to the porous medium itself,
�p ¼ Ap� ðA:7Þ
where Ap is the total strain concentration factor for the
inclusions(pores). This factor can be obtained in the form,
Ap ¼ fTpgðnfTpg þ ð1� nÞT sÞ�1 ðA:8Þ
Eq. (A.7) thus gives the orientation averaged strain in
theinclusions.
The overall stiffness matrix of the porous medium can be
foundas,
L ¼ Ls þ nðLp � LsÞAp ðA:9Þ
If the properties of a drained porous medium are sought, the
stiff-ness matrix of the pores is zero Lp = 0. On the other hand,
for un-drained conditions, Lp is a regular stiffness matrix that is
obtainedby setting the shear modulus Gf of the fluid in the pore to
zero.
Formal equivalence of all effective medium methods allows usto
find the self-consistent estimate iteratively. Starting with
somecomparison medium Lð1Þ0 , possibly different from the solid
phase Ls,we generate, using (A.5), (A.6) and (A.9), the estimate of
the overallmoduli L(1). At the next iteration we use this estimate
as the newcomparison medium L0, i.e., L
ð2Þ0 ¼ L
ð1Þ. Every time, the overall stiff-ness matrix obtained at a
previous iteration serves as a comparisonmedium at the subsequent
iteration, i.e., Lðiþ1Þ0 ¼ L
ðiÞ. Iterations areterminated when the difference between the
stiffness of the com-parison medium LðiÞ0 and the overall stiffness
matrix L
(i) is small.Now we prove that the Mori–Tanaka estimates of the
bulk mod-
ulus satisfy the Biot–Gassmann equation. For an arbitrary matrix
Awe define the volumetric part of the matrix as½A�V ¼ 13 ðA11 þ A22
þ A33 þ A12 þ A21 þ A13 þ A31 þ A23 þ A32Þ. First of
all note that from (A.9) the bulk modulus of the undrained
mediumKu can be represented as,
Ku ¼ Ks þ nAof ðKf � KsÞ ðA:10Þ
where Aof is the volumetric strain concentration factor under
un-drained conditions. For the drained porous medium,
KD ¼ Ks þ nAof ð�KsÞ ðA:11Þ
where Aof is the volumetric part of the strain concentration
factorunder drained conditions. If the Biot–Gassmann Eq. (3) is
satisfied,then it can be shown that,
Af ¼a
nþ ða� nÞKf =Ks¼
Aof1þ ðAof � 1ÞKf =Ks
ðA:12Þ
where Aof ¼ a=n, and a = (1 � KD/Ks) is the Biot coefficient.
Thenfrom (A.12),
1Af¼ 1
Aofþ 1� 1
Aof
!KfKs
ðA:13Þ
In the Mori–Tanaka method, the volumetric concentration
factorsAf, A
of can be found from (A.8) in terms of partial volumetric
concen-
tration factors as follows:
Aof ¼ ðnþ ð1� nÞðTof Þ�1Þ�1; Af ¼ ðnþ ð1� nÞðTf Þ�1Þ�1
ðA:14Þ
where Tf and Tof are partial volumetric concentration factors
for the
undrained and drained porous media, respectively. They are
definedas volumetric parts of the averaged matrices, i.e., Tf =
[{Tp}]V,Tof ¼ ½fT
opg�V .
Partial volumetric concentration factors can be expressed
from(A.2) as,
Tf ¼ ð1� PV Ks þ PV Kf Þ�1 Tof ¼ ð1� PV KsÞ�1 ðA:15Þ
where PV is a certain factor related to the Hill’s polarization
tensor P.PV depends on the shape of the pore and the elastic moduli
of thematrix material, i.e., solid phase, but not on the moduli of
the pores.PV is unique both for Tf and T
of , and can be expressed in terms of T
of
as PV ¼ ð1� Tof� ��1
ÞK�1s . It is important to note that (A.14) is validonly for the
Mori–Tanaka method, but the formula (A.15) is validfor any choice
of comparison material once Ks is replaced with thebulk modulus of
that material.
Now after substitution of (A.15) into (A.14), the equality
(A.13)or (A.12) can be proved. Consequently, the Biot–Gassmann Eq.
(3)is satisfied.
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Effective medium methods and a computational approach for
estimating geomaterial properties of porous materials with randomly
oriented ellipsoidal pores1 Introduction2 Estimates of overall
elastic moduli and their properties3 Numerical results and
modelling4 New Biot-consistent estimates of elastic moduli5
Estimate for permeability6 ConclusionsAcknowledgementAppendix A
References